First, "well done" to all involved. Second, thank you for posting this experience for us to learn from. Third, as to consent to treat, generally, state statutes authorize any necessary emergency treatment if no parent is available. YMMV.

More minor or routine care is actually more problematic. Babysitters who are themselves minors may have a bit more trouble, as well. The consent forms shown aren't bad, but notarization is the gold standard. The form itself should mention both emergency and routine care authority, if, for example, you might need your babysitter to take your kid in for a dentist's appointment when you're running late. It should also state your acceptance of financial responsibility, and include your own full name, SSN, DOB, address, etc., as well as your child's. It should also have at least two adult relatives' and your pediatrician's full contact information.

On a separate page, include a full medical history, meds, and allergies, since these things do change. Then, attach a photocopy of your DL or other ID and your insurance card.

Does your babysitter know infant/child CPR and first aid? Where your fire extinguisher is and how to use it? Your address and telephone numbers by memory? Who to call for help if you can't be reached for some reason, other than 911?